Table 10.

Convergence results for the inversion using our formulation for a fully incompressible diffeomorphism (linear Stokes regularization). We report results for registering the UT images (see Fig. 3; grid size: 256 × 256) using our original preconditioner (P_REG; RK2A scheme with a CFL number of 0.2) and the proposed preconditioner (P_2L; SL scheme with a CFL number of 5; CHEB method with a fixed number of 10 iterations). We consider different regularization norms: an H¹-seminorm; an H²-seminorm; and an H³-seminorm (from top to bottom). We terminate the inversion if the change in the ℓ^∞-norm of the reduced gradient ${\mathit{g}}_k^h$, k = 1, 2, …, is at least two orders of magnitude or if the ℓ^∞-norm of ${\mathit{g}}_k^h$ is smaller or equal to 1E−5. We report (i) the relative change of the reduced gradient ‖g^★‖_rel, (ii) the relative change of the residual ‖r‖ rel (L²-distance between m_R and m₁), (iii) the number of outer iterations, (iv) the number of Hessian matvecs, (v) the time to solution, and (vi) the speedup compared to our original scheme.

run	norm	P	PDE solver	PC solver	‖g★‖rel	‖r‖rel	iter	matvecs	time	speedup
#1	H1	PREG	RK2A(0.2)	—	8.04E−3	1.40E−2	12	137	1.20E+3	—
#2		P2L	SL(5)	CHEB(10)	6.40E−3	1.38E−2	13	43	8.88E+1	1.35E+1
#3	H2	PREG	RK2A(0.2)	—	9.05E−3	1.90E−1	17	177	1.38E+3	—
#4		P2L	SL(5)	CHEB(10)	8.81E−3	1.90E−1	16	53	1.06E+2	1.30E+1
#5	H3	PREG	RK2A(0.2)	—	9.14E−3	6.24E−1	34	402	2.60E+3	—
#6		P2L	SL(5)	CHEB(10)	9.30E−3	6.24E−1	36	111	2.22E+2	1.17E+1